为实现大规模电力系统潮流的准确、快速求解,以非精确牛顿法为基础,提出一种基于CPU-GPU异构平台的电力系统潮流并行计算方法。修正方程组的求解是牛拉法潮流计算中最为耗时的部分,提升修正方程组的求解效率可有效提升潮流计算效率。为...为实现大规模电力系统潮流的准确、快速求解,以非精确牛顿法为基础,提出一种基于CPU-GPU异构平台的电力系统潮流并行计算方法。修正方程组的求解是牛拉法潮流计算中最为耗时的部分,提升修正方程组的求解效率可有效提升潮流计算效率。为此,根据雅可比矩阵的不对称不定性,采用稳定双正交共轭梯度(bi-conjugate gradient stabilized,BICGSTAB)法进行修正方程组的求解。进一步,为改善BICGSTAB法的收敛性,根据雅可比矩阵的稀疏性和类对角占优性,提出一种改进PPAT(Preconditioner with sparsity Pattern of AT,PPAT)预处理器和改进Jacobi预处理器相结合的两阶段预处理方法,并对雅可比矩阵进行预处理,提升BICGSTAB法的收敛性能。然后,将上述潮流算法移植到CPU-GPU异构平台,实现电力系统潮流的并行求解。最后,通过不同测试系统算例对所提方法进行验证、分析。结果表明,所提潮流并行计算方法可实现电力系统潮流的准确、快速求解。展开更多
With the aid of index functions,we re-derive the ML(n)BiCGStab algorithm in[Yeung and Chan,SIAM J.Sci.Comput.,21(1999),pp.1263-1290]systematically.There are n ways to define the ML(n)BiCGStab residual vector.Each defi...With the aid of index functions,we re-derive the ML(n)BiCGStab algorithm in[Yeung and Chan,SIAM J.Sci.Comput.,21(1999),pp.1263-1290]systematically.There are n ways to define the ML(n)BiCGStab residual vector.Each definition leads to a different ML(n)BiCGStab algorithm.We demonstrate this by presenting a second algorithm which requires less storage.In theory,this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML(n)BiCG is a bridge connecting BiCG and FOM.We also analyze the breakdown situation from the probabilistic point of view and summarize some useful properties of ML(n)BiCGStab.Implementation issues are also addressed.展开更多
The reliability of BiCGStab and IDR solvers for the exponential scheme discretization of the advection-diffusion-reaction equation is investigated.The resulting discretization matrices have real eigenvalues.We conside...The reliability of BiCGStab and IDR solvers for the exponential scheme discretization of the advection-diffusion-reaction equation is investigated.The resulting discretization matrices have real eigenvalues.We consider BiCGStab,IDR(S),BiCGStab(L)and various modifications of BiCGStab,where S denotes the dimension of the shadow space and L the degree of the polynomial used in the polynomial part.Several implementations of BiCGStab exist which are equivalent in exact arithmetic,however,not in finite precision arithmetic.The modifications of BiCGStab we consider are;choosing a random shadow vector,a reliable updating scheme,and storing the best intermediate solution.It is shown that the Local Minimal Residual algorithm,a method similar to the“minimize residual”step of BiCGStab,can be interpreted in terms of a time-dependent advection-diffusion-reaction equation with homogeneous Dirichlet boundary conditions for the residual,which plays a key role in the convergence analysis.Due to the real eigenvalues,the benefit of BiCGStab(L)compared to BiCGStab is shown to be modest in numerical experiments.Non-sparse(e.g.uniform random)shadow residual turns out to be essential for the reliability of BiCGStab.The reliable updating scheme ensures the required tolerance is truly achieved.Keeping the best intermediate solution has no significant effect.Recommendation is to modify BiCGStab with a random shadow residual and the reliable updating scheme,especially in the regime of large P´eclet and small Damk¨ohler numbers.An alternative option is IDR(S),which outperforms BiCGStab for problems with strong advection in terms of the number of matrix-vector products.The MATLAB code used in the numerical experiments is available on GitLab:https://gitlab.com/ChrisSchoutrop/krylov-adr,a C++implementation of IDR(S)is available in the Eigen linear algebra library:http://eigen.tuxfamily.org.展开更多
文摘为实现大规模电力系统潮流的准确、快速求解,以非精确牛顿法为基础,提出一种基于CPU-GPU异构平台的电力系统潮流并行计算方法。修正方程组的求解是牛拉法潮流计算中最为耗时的部分,提升修正方程组的求解效率可有效提升潮流计算效率。为此,根据雅可比矩阵的不对称不定性,采用稳定双正交共轭梯度(bi-conjugate gradient stabilized,BICGSTAB)法进行修正方程组的求解。进一步,为改善BICGSTAB法的收敛性,根据雅可比矩阵的稀疏性和类对角占优性,提出一种改进PPAT(Preconditioner with sparsity Pattern of AT,PPAT)预处理器和改进Jacobi预处理器相结合的两阶段预处理方法,并对雅可比矩阵进行预处理,提升BICGSTAB法的收敛性能。然后,将上述潮流算法移植到CPU-GPU异构平台,实现电力系统潮流的并行求解。最后,通过不同测试系统算例对所提方法进行验证、分析。结果表明,所提潮流并行计算方法可实现电力系统潮流的准确、快速求解。
基金Project(42274083) supported by the National Natural Science Foundation of ChinaProject(2023JJ30659) supported by Hunan Provincial Natural Science Foundation of China。
基金supported by 2008 Flittie Sabbatical Augmentation Award,University of Wyoming.
文摘With the aid of index functions,we re-derive the ML(n)BiCGStab algorithm in[Yeung and Chan,SIAM J.Sci.Comput.,21(1999),pp.1263-1290]systematically.There are n ways to define the ML(n)BiCGStab residual vector.Each definition leads to a different ML(n)BiCGStab algorithm.We demonstrate this by presenting a second algorithm which requires less storage.In theory,this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML(n)BiCG is a bridge connecting BiCG and FOM.We also analyze the breakdown situation from the probabilistic point of view and summarize some useful properties of ML(n)BiCGStab.Implementation issues are also addressed.
文摘The reliability of BiCGStab and IDR solvers for the exponential scheme discretization of the advection-diffusion-reaction equation is investigated.The resulting discretization matrices have real eigenvalues.We consider BiCGStab,IDR(S),BiCGStab(L)and various modifications of BiCGStab,where S denotes the dimension of the shadow space and L the degree of the polynomial used in the polynomial part.Several implementations of BiCGStab exist which are equivalent in exact arithmetic,however,not in finite precision arithmetic.The modifications of BiCGStab we consider are;choosing a random shadow vector,a reliable updating scheme,and storing the best intermediate solution.It is shown that the Local Minimal Residual algorithm,a method similar to the“minimize residual”step of BiCGStab,can be interpreted in terms of a time-dependent advection-diffusion-reaction equation with homogeneous Dirichlet boundary conditions for the residual,which plays a key role in the convergence analysis.Due to the real eigenvalues,the benefit of BiCGStab(L)compared to BiCGStab is shown to be modest in numerical experiments.Non-sparse(e.g.uniform random)shadow residual turns out to be essential for the reliability of BiCGStab.The reliable updating scheme ensures the required tolerance is truly achieved.Keeping the best intermediate solution has no significant effect.Recommendation is to modify BiCGStab with a random shadow residual and the reliable updating scheme,especially in the regime of large P´eclet and small Damk¨ohler numbers.An alternative option is IDR(S),which outperforms BiCGStab for problems with strong advection in terms of the number of matrix-vector products.The MATLAB code used in the numerical experiments is available on GitLab:https://gitlab.com/ChrisSchoutrop/krylov-adr,a C++implementation of IDR(S)is available in the Eigen linear algebra library:http://eigen.tuxfamily.org.
